Theoretical study of the stretched-pulse erbium-doped fiber laser

Theoretical study of the stretched-pulse erbium-doped fiber laser

M. Salhi,*H. Leblond, and F. Sanchez

Laboratoire des POMA, Universite´ d’Angers, 2 Boulevard Lavoisier, 49045 Angers cedex, France 共Received 28 February 2003; revised manuscript received 7 July 2003; published 29 September 2003兲 The mode-locking properties of a stretched-pulse erbium-doped fiber laser are theoretically investigated. We consider a unidirectional ring cavity composed of a birefringent erbium-doped fiber and a birefringent undoped fiber. An intracavity polarizer is used to achieve mode locking through nonlinear polarization rotation. The model takes into account the orientation of the eigenaxes of both fibers as well as the orientation of the intracavity polarizer. A master mode-locking equation is derived for the envelope of the electric field. This allows an analytic investigation of the stability of the mode-locking solution as a function of the angle between the eigenaxes of the two fibers. It is demonstrated that the optimization of the ranges of mode locking requires the alignment of the eigenaxis of the two fibers.

DOI: 10.1103/PhysRevA.68.033815 PACS number共s兲: 42.55.Wd

I. INTRODUCTION

The stretched-pulse regime has been extensively investi- gated both experimentally and theoretically for the realiza- tion of passively mode-locked fiber lasers. The principle of the method consists in splicing two pieces of fiber having chromatic dispersion with opposite signs. The total cavity dispersion is slightly positive. One of the fibers is doped with rare-earth ions while the other is a passive fiber. The fiber with positive chromatic dispersion can be replaced by a grat- ing pair or a prism pair. The fibers are then mounted in a unidirectional ring cavity containing a polarizing isolator placed between two polarization controllers. This allows one to passively mode lock the fiber laser 关1–3兴. The physics responsible for the mode-locking effect is the nonlinear po- larization rotation resulting from self-phase and cross-phase modulations关4兴. A short pulse propagating in such a cavity is stretched and then compressed. This allows one to limit the peak power, thus ensuring both a limitation of the nonlinear phase shift per round-trip and the non-saturation of the non- linear polarization rotation. Stretched-pulse mode locking is then a powerful method to generate high-energy ultrashort pulses 关5,6兴. In part, this is due to the fact that sideband generation is eliminated because the total cavity dispersion is slightly positive. In the opposite case when the total cavity dispersion is negative, the laser operates in the soliton re- gime关7兴. Let us finally note that the point where the power is extracted from the cavity is important because the pulses are short in a small portion of the cavity only.

Experimentally, the method of stretched-pulse mode- locking has been successfully used to generate pulses of about 70 fs with an energy of 90 pJ after compensation of the chirp关1,8兴. On the same time, theoretical models have been developed 关1,9,10兴. The models are based on the master equation proposed by Haus关9兴, which takes into account the variation of the chromatic dispersion along the cavity. Al- though this model qualitatively describes the mode-locking properties of the laser, it does not allow one to investigate the influence of the angle between the eigenaxes of the two fi- bers which generally are birefringent.

The aim of this paper is to investigate theoretically an erbium-doped fiber laser operating in the stretched-pulse re- gime. We consider a unidirectional ring cavity composed of a birefringent erbium-doped fiber with positive group velocity dispersion共GVD兲and a passive birefringent fiber with nega- tive GVD. A schematic representation of the system is shown in Fig. 1. The cavity contains a polarizing isolator placed between two half-wave plates. In the general case, two po- larization controllers are used instead of two half wave plates. We consider here a simpler situation in order to re- duce the number of degrees of freedom to one per plate instead of two for a polarization controller. In contrast with previous works, the angle ␣ between the eigenaxes of the fibers is explicitly taken into account and considered as vari- able. Our model is based on a master equation which takes into account the angles between the eigenaxes of the fiber and the passing axis of the polarizer关11兴. We will denote by

␪_⫺⁽␪_⫹) the angle between the eigenaxes of the fiber and the passing axis of the polarizer before共after兲the polarizer. This model has been successfully applied to the case of the ytterbium-doped fiber laser operating in the polarization ad- ditive pulse mode-locking regime 关12兴. It has been then adapted to the case of the erbium-doped fiber laser operating in the soliton regime关13兴. In particular, we have optimized both the pulse duration and energy versus the orientation of the polarizer. The paper is organized as follows. In Sec. II we derive the master equation for the stretched-pulse laser. The resulting nonlinear propagation equation, which is of com-

*Electronic address: mohamed.salhi@univ-angers.fr FIG. 1. Setup for a stretched-pulse mode-locking laser.

plex Ginzburg-Landau 共CGL兲type, explicitly takes into ac- count the angles␣^,␪_⫺^{, and}␪_⫹. Two particular solutions of the CGL equation are considered in Sec. III: the constant solution, which corresponds to a continuous wave 共cw兲 op- eration of the laser, and the explicit localized solution, which corresponds to the mode-locked regime of the laser. In Sec.

IV we first investigate the influence of␣^,␪_⫺^{, and}␪_⫹^{on the} mode-locking properties of the laser.

II. DERIVATION OF THE MASTER EQUATION In this section we derive a master equation for the cavity illustrated in Fig. 1. The numerical values of the parameters used for the simulations are as follows: the erbium-doped fiber has a length L⫽9 m, a birefringent parameter K

⫽0.1 m^⫺1, a nonlinear coefficient␥⫽0.002 W^⫺1m^⫺1, and a GVD␤2⫽0.075 ps²m^⫺1. The undoped fiber is character- ized by its variable length L

⬘

, a birefringent parameter K

⬘

⫽2 m^⫺1, a nonlinear coefficient␥

⬘

⫽0.002 W^⫺1m^⫺1, and a GVD ␤2

⬘

⫽⫺0.022 ps²m^⫺1.

A. Propagation along the fibers

The envelope of the electric field is first decomposed on the basis of the eigenaxes of the erbium-doped fiber. Each component is affected by the optical Kerr effect, the GVD, the birefringence, and the gain. In the framework of the ei- genaxes of the fiber propagating at the group velocity, the propagation is modeled through two coupled nonlinear Schro¨dinger equations关2,11,13–15兴:

i⳵zu⫺Ku⫺␤2

2 ⳵t

2u⫹␥共u兩u兩²⫹Au兩v兩²⫹Bv²u*_兲

⫽ig

冉

^{1⫹␻1g2⳵t2}

冊

^{u, 共1兲}

i⳵zv⫹Kv⫺␤2

2 ⳵t

2u⫹␥共v兩v兩²⫹Av兩u兩²⫹Bu²v*_兲

⫽ig

冉

^{1⫹␻1g2⳵t2}

冊

^{v, 共2兲}

where⳵t stands for the partial derivative ⳵^/⳵^{t, g in m⫺1 is} the linear gain,␻g⫽15.7 ps^⫺1is the spectral gain linewidth, A⫽2/3, and B⫽1/3. The values of A and B are determined by the material; we consider silica fibers.

Let us now briefly recall the procedure we use to solve the problem 关11,13兴. Equations 共1兲 and 共2兲 are solved using a perturbative approach. We assume that the effect of the GVD

␤2, the nonlinear coefficient ␥, and the gain filtering ␳

⫽g/␻g

2 are small over one round-trip of the cavity. This allows us to treat these quantities as perturbations. We intro- duce therefore a small parameter ␧ and replace ␤2, ␥^{, and}

␳^{, by}␧␤2,␧␥^{, and}␧␳, respectively. Using this perturbative approach, we compute the field components„u_⫺(L),v_⫺(L)… after propagation over a distance L, at the output of the fiber,

as functions of the electric-field components at the entrance of the erbium-doped fiber„u(0),v(0)… 关Eqs.共A1兲 and共A2兲 in the Appendix兴.

The erbium-doped fiber is spliced with an undoped fiber.

In the general case, the eigenaxes of the fibers are not aligned. Denote by␣ the angle between the eigenaxes of the two fibers. The field components at the entrance of the un- doped fiber „u_⫹(L),v_⫹(L)… are linked to „u_⫺(L),v_⫺(L)… through the rotation:

u_⫹共L兲⫽u_⫺共L兲cos␣⫹v_⫺共L兲sin␣^, 共3兲 v_⫹共L兲⫽⫺u_⫺共L兲sin␣⫹v_⫺共L兲cos␣^. 共4兲 Propagation of the field along the undoped fiber is modeled with two nonlinear Schro¨dinger equations similar to Eqs.共1兲 and共2兲but without gain,

i⳵zu⫺K

⬘

^u⫺␤2

⬘

2 ⳵t

2u⫹␥

⬘

共u兩u兩²⫹Au兩v兩²⫹Bv²u*_兲⫽0, 共5兲

i⳵zv⫹K

⬘

^v⫺␤2

⬘

2 ⳵t

2u⫹␥

⬘

共v兩v兩²⫹Av兩u兩²⫹Bu²v*_兲⫽0.

共6兲 Equations 共5兲 and 共6兲 are solved using a perturbative ap- proach which assumes that the effects of␥

⬘

^and␤2

⬘

^{are small} over one round-trip of the cavity. This allows us to find the electric-field components at the exit of the undoped fiber

„u(L⫹L

⬘

^),v(L⫹L

⬘

⁾… 关Eqs. 共A3兲 and 共A4兲 in the Appen- dix兴.

B. Polarization conditions

The quantities u_⫹(L), v_⫹(L) in Eqs.共A3兲 and共A4兲are replaced by the expressions 共3兲 and 共4兲. Then u_⫺(L) and v_⫺(L) are replaced using the formulas共A1兲and共A2兲. This procedure allows to obtain the evolution of the field along the fibered part of the cavity. It remains to take into account the effect of the polarizer. Because of the linear birefringence and the induced birefringence from the optical Kerr effect, the polarization state of the electric field at the exit of the undoped fiber is elliptic. The effect of the polarizer is to project the electric field along its passing axis. After the po- larizer, the field is then linearly polarized. After the nth round-trip, the electric-field components at the entrance of the doped fiber are

冉

^{uvnn共共00兲兲}

冊

^⫽

冉

^cossin␪␪⫹^⫹

冊

^{fn, 共7兲}

where f_n is the electric-field amplitude after the nth round trip, and ␪_⫹ is the angle between the eigenaxes of the erbium-doped fiber and the passing axis of the polarizer. Fig- ure 2 gives the definition of the angles of interest for our study.

At the exit of the undoped fiber, the eigenaxes corre- sponding to the component u makes an angle ␪_⫺ ^{with the} passing axis of the polarizer, as shown on Fig. 2. The polar-

izer is characterized by an amplitude transmission coefficient

␤⫽95%. It lets pass only light polarized along its passing axis. The electric-field amplitude after the (n⫹1)th round trip, just after the polarizer, expresses as a function of the electric-field components„u_n(L⫹L

⬘

^),vn(L⫹L

⬘

⁾…at the exit of the undoped fiber as

f_n⫹1⫽␤共cos␪_⫺^,sin␪_⫺兲

冉

^{uvnn共共LL⫹⫹LL}

^{⬘ ⬘}

^兲兲

冊

^{. 共8兲}

C. The master equation for the stretched-pulse laser We set ␹⫽␳^{L and} ␩⫽␤2L⫹␤2

⬘

^L

⬘

, which is the total cavity dispersion. The field amplitude at the (n⫹1)th round- trip just after the polarizer is expressed as a function of its amplitude at the nth round-trip as follows. First we replace

„u_n(L⫹L

⬘

^),vn(L⫹L

⬘

⁾…in Eq.共8兲by their expressions共A3兲 and共A4兲, and„u_⫹,n(L),v_⫹,n(L)…using relations共3兲and共4兲. Then we make use of expressions 共A1兲 and Eq. 共A2兲 of

„u_⫺,n(L),v_⫺,n(L)… and共7兲of„u_n(0),vn(0)…, to obtain f_n⫹1⫽␤^{egLQ f}n⫹␧␤^egL

冋冉

^␹⫺2i␩

冊

^{Q⳵t2fn⫹i P fn兩fn兩2}

册

⫹O共␧²兲, 共9兲 where the complex coefficients Q and P are given in the Appendix. Their expressions involve the various parameters of the problem, in particular, they depend explicitly on the angles ␪_⫺^, ␪_⫹^{, and}␣^.

The gain threshold g₀ is determined by requiring that the considered state be stationary, thus that 兩f_n⫹1兩⫽兩f_n兩. Ac- cording to expression 共9兲of f_n⫹1 at lowest order ␧⁰, at sta- tionary state the modulus of ␤^egLQ is 1. This yields the expression of g₀ as

g₀⫽⫺1

2L ln共␤²兩Q兩²兲, 共10兲 where 兩Q兩² is given by formula 共A17兲. Since Eq. 共10兲 is a first-order approximation, we write the gain g as

g⫽g₀⫹␧g₁, 共11兲 where g₁ is the excess of linear gain.

Performing a Taylor expansion of e^␧g1L, and replacing

␤^{eg0LQ by ei␺, Eq.} 共9兲becomes

f_n⫹1⫽e^i␺共1⫹␧g₁L兲f_n⫹␧

冉

^␹⫺2i␩

冊

^ei␺⳵t2fn

⫹i␧e^i␺

Q P f_n兩f_n兩²⫹O共␧²兲. 共12兲

Equation 共12兲 is a discrete equation where the variable n represents the number of round trips in the cavity. It is more convenient to write an equation for a continuous variable.

This is done by interpolating the discrete sequence f_n by a continuous function f„z⫽n(L⫹L

⬘

⁾…. We obtain

i⳵zf⫽

冉

^L⫺⫹␺L

^⬘

^{⫹i␧Lg⫹1LL}

^⬘ 冊

^f⫹␧12

冉

^{␩L⫹⫹2iL}

^⬘

^␹

冊

^{⳵t2f⫹␧Df兩f兩2,}

共13兲 where

D⫽ ⫺P

Q共L⫹L

⬘

兲. 共14兲

For a large number of round trips, i.e., for long propagation distances z, we introduce a slow variable␨⫽␧z. A multiscale analysis关11兴allows us to convert the correction of order␧ in Eq. 共13兲into the long-distance evolution equation

i⳵_␨^F⫽i g₁L L⫹L

⬘

^F⫹

1

2

冉

^{␩L⫹⫹2iL}

^⬘

^␹

冊

^{⳵t2F⫹共Dr⫹iDi兲F兩F兩2.}

共15兲 The amplitude F is related to f through

f⫽Fe^{i␺z/(L⫹L⬘)}⫹O共␧兲, 共16兲

which is the solution of Eq.共13兲at order ␧⁰.

Equation共15兲is the sought master equation of cubic CGL type. It is formally identical to the master equation proposed by Haus关18兴.Dr andDi are the real and imaginary parts of D. They correspond, respectively, to the effective self-phase modulation and to the effective nonlinear gain or absorption.

The latter is due to the combined effects of the nonlinear polarization rotation, the polarizer, and the linear gain. The sign of Di depends on the angles ␪_⫺^, ␪_⫹^{, and}␣^.

III. SOLUTIONS OF THE CGL EQUATION In this section we are interested in two particular solutions of the CGL equation. The first one is the constant amplitude solution which corresponds to a continuous-wave operation of the laser. The second is the explicit localized solution, which corresponds to the mode-locked operation of the laser.

In both cases, stability conditions are analyzed and dis- cussed.

FIG. 2. Definition of the angles␪_⫺and␪_⫹.

A. Constant solution

The CGL equation共15兲admits the following constant so- lution:

F⫽Ae^{i(k␨⫺⍀t)}, 共17兲 where

⍀²⫽L⫹L

⬘

␹

冉

^{Di兩A兩2⫹g1L⫹LL}

^⬘ 冊

^{, 共18兲}

and

k⫽ ␩

2␹

冉

^{Di兩A兩2⫹g1L⫹LL}

^⬘ 冊

^{⫺Dr兩A兩2. 共19兲}

The solution given by Eq.共17兲has a constant amplitude. It is independent of time when ⍀⫽0. In that case, the expres- sions of Aand k are

A⫽

冑

Di^⫺共L^g⫹^1LL

⬘

兲, 共20兲 k⫽ DrL

Di共L⫹L

⬘

兲g₁. 共21兲 This solution exists only if the product Dig₁ is negative.

Moreover, it has been shown that the modulational instability occurs when the excess of linear gain g₁ is negative and the nonlinear gain Di is positive 关11兴. Therefore the solution with constant amplitude is stable when g₁ is positive andDi

is negative.

B. Localized solution

Equation共33兲also admits an explicit localized solution of the form 关16兴

F⫽a共t兲^1⫹ide^⫺i␻␨, 共22兲 where

d⫽⫺3关␩Dr⫹2␹Di兴⫹

冑

⁹关2␹Di⫹␩Dr兴²⫹8关␩Di⫺2␹Dr兴²

2关␩Di⫺2␹Dr兴 , 共23兲

and

␻⫽⫺g₁L关4␹^d⫹␩^d2⫺␩兴

2关␩^d2⫺␹⫺␩^d兴 . 共24兲 The real amplitude a(t) is

a共t兲⫽M N sech共M t兲, 共25兲 where

M⫽

冑

_␹d²⫺^g1␹^L⫺␩^{d, 共26兲} N⫽

冑

2共L⫹^3dL^关

⬘

兲关^4␹␩²D^⫹i⫺^␩22^兴␹Dr兴, 共27兲

d is the chirp parameter.

This analytical solution is localized if M is real, i.e., if the excess of linear gain g₁ and the quantity (␹^d2⫺␹⫺␩^d) have the same sign. In the case of normal dispersion (␩

⬎0), this solution is never stable 关17兴. However, it can be stabilized when considering higher-order terms, namely, a quintic nonlinear term关17兴. Such a ‘‘potential stability’’ oc- curs when the zero solution is unstable, i.e., when the excess of linear gain g₁is positive. We can thus use, as in Ref.关11兴, the following stability criterion:

␹^d2⫺␹⫺␩^{d⬍0. 共28兲}

Its physical relevance has been verified experimentally 关11兴. However, the stability of solution 共22兲has been considered only in the case where the nonlinear gainDi is positive. In the opposite case, since the constant nonzero amplitude so- FIG. 3. Stability regions of the cw and the mode-locked regimes in the plane (␪_⫺,␪_⫹) for a total cavity dispersion of 0.037 ps²and for␣⫽0°. In the white region, the cw regime is stable, in the light gray region both the cw and the mode-locked regimes are unstable;

and in the dark gray region the mode-locked regime is stable while the cw regime is unstable.

lution is then stable, we will be able to conclude about laser operation. Situations in which both cw operation and mode locking can be observed for the same position of the half wave plates have been observed. The theoretical study of these situations involves the knowledge of the stability con- dition of the localized solution forDi⬍0. It is left for further investigation.

IV. INFLUENCE OF␣ON THE OPERATING REGIME OF THE LASER

In Sec. III we have given the stability criterion for the continuous and mode-locked regimes. The stability depends on the angles ␪_⫺^, ␪_⫹^{, and} ␣ through the values of the coefficientsDr, Di, and of the first-order gain g₀. We now perform a systematic study of the mode-locking regions as a

function of ␣, using the expressions of Dr, Di, and g₀ de- rived in Sec. II and the stability criteria recalled in Sec. III.

For that we vary␣by step of 5° and look for the regions of stability of the two solutions considered in this paper in the plane (␪_⫺^,␪_⫹). Indeed, such a mapping is a very convenient representation of the results. The results are periodic with a periodicity of 180° versus all angles. The total cavity disper- sion ␩ has been first fixed to 0.037 ps². Numerical compu- tations show that the mode-locking regions have maximal area for ␣⫽0°, ␣⫽90°, and ␣⫽180°. Conversely, these regions are very small in the vicinity of ␣⫽45° and ␣

⫽135°. Hence, the mode-locking regime is more efficient when the eigenaxes of the doped and the undoped fibers are roughly aligned, i.e. when␣⫽0° or␣⫽90°, which is rather intuitive. These results are illustrated in Figs. 3 and 4 which give the regions of stability of both the cw and the mode- locked regimes for␣⫽0° and ␣⫽135°, respectively.

We have performed the same study for a total cavity dis- persion ␩⫽0.011 ps². We observed again a strong depen- dence of the mode-locking regions upon ␣. Globally, the mode locking occurs within smaller regions than for the pre- vious value of the GVD. This somehow paradoxical feature has been found from a different approach by Haus et al.关18兴. Notice that the stability domains have a different shape. The dependency with regard to the angle␣ is mainly the same as the previous: the domains of mode locking are maximal when the eigenaxes of the two pieces of fiber are roughly aligned, i.e., when ␣ is small or close to 90°, as shown in Figs. 5 and 6. The domains of mode locking vanish when

␣ is about 45° or 135°. However, there is some special behavior for exact alignment of the fiber eigenaxes, with both fast axes and both slow axes accurately aligned, i.e., when ␣⫽0. In this case the mode-locking domains vanish completely. The corresponding cartography is not shown since it is very similar to Fig. 4. Due to this unexpected effect, the mode-locking domains are optimized for ␣⯝5°

共Fig. 5兲. FIG. 4. Same as Fig. 3 except that␣⫽135°共total cavity disper-

sion:␩⫽0.037 ps²).

FIG. 5. Same as Fig. 3 except that the total cavity dispersion is 0.011 ps², and the angle␣ is 5°.

FIG. 6. Same as Fig. 5 except that␣⫽89.5° 共total cavity dis- persion:␩⫽0.011 ps²).

The dependency of the mode locking with regard to the GVD will be investigated more precisely in the future.

V. CONCLUSION

In conclusion, we have established a master equation for a laser operating in the stretched-pulse mode-locking regime.

The model takes into account the angle ␣ between the eigenaxes of the doped and undoped fibers. The resulting equation is of the CGL type. It admits a nonzero constant solution, which corresponds to a continuous-wave regime, and an explicit localized solution, which corresponds to the mode locked regime. Investigation of the stability of the lo- calized solution has allowed us to identify the influence of␣ on the area of the domains of mode locking. We have shown that the best configuration is obtained when the eigenaxes of the fibers are aligned. This is an important result for the design of practical stretched-pulse mode-locking lasers. In- deed, even if standard fibers are generally slightly birefrin- gent, rare-earth doped fibers have a beating length of the order of 1 m 关19兴. Therefore, the birefringence has to be taken into account when realizing a stretched-pulse laser.

APPENDIX: DETAIL ON THE DERIVATION OF THE MASTER EQUATION 1. Resolution of the propagation equations

Equations共1兲and共2兲govern the evolution of the electric- field components (u,v) along the erbium-doped fiber. Recall that we denote by„u(0),v(0)…the electric-field components at the entrance of the erbium-doped fiber and by

„u_⫺(L),v_⫺(L)…the field components after propagation over a distance L, at the output of the fiber. Resolution of Eqs.共1兲 and共2兲using a perturbative method yields

u_⫺共L兲⫽u共0兲e^(g⫺iK)L⫹␧

冋

^L

冉

^␳⫺i␤22

冊

^{⳵t2u共0兲}

⫹i␥关u共0兲兩u共0兲兩²⫹Au共0兲兩v共0兲兩²兴e^2gL⫺1 2g

⫹i␥^Bv共0兲²u共0兲*^e

(2g⫹4iK)L⫺1

2g⫹4iK

册

^e(g⫺iK)L

⫹O共␧²兲, 共A1兲

v_⫺共L兲⫽v共0兲e^(g⫹iK)L⫹␧

冋

^L

冉

^␳⫺i␤22

冊

^{⳵t2v共0兲}

⫹i␥关v共0兲兩v共0兲兩²⫹Av共0兲兩u共0兲兩²兴e^2gL⫺1 2g

⫹i␥^Bu共0兲²v共0兲*^e

(2g⫺4iK)L⫺1

2g⫺4iK

册

^e(g⫹iK)L

⫹O共␧²兲. 共A2兲

Equations 共5兲 and 共6兲 govern the propagation along the undoped fiber. They are solved using a perturbative approach which allows us to find the electric-field components at the exit of the undoped fiber „u(L⫹L

⬘

^),v(L⫹L

⬘

⁾…, as

u共L⫹L

⬘

兲⫽u_⫹共L兲e^⫺iK⬘L⬘⫹␧

冋

^⫺i␤22

^⬘

^L

^⬘

^{⳵t2u⫹共L兲}

⫹i␥

⬘

^L

⬘

关u_⫹共L兲兩u_⫹共L兲兩²⫹Au_⫹共L兲兩v_⫹共L兲兩²兴

⫹␥

⬘

^B 4K

⬘

^共e

4iK⬘^L⬘⫺1兲v_⫹共L兲²u_⫹共L兲*

册

^{e⫺iK⬘L⬘}

⫹O共␧²兲, 共A3兲

v共L⫹L

⬘

^兲⫽v⫹共L兲e^iK⬘L⬘⫹␧

冋

^⫺i␤22

^⬘

^L

^⬘

^{⳵t2v⫹共L兲}

⫹i␥

⬘

^L

⬘

关v_⫹共L兲兩v_⫹共L兲兩²⫹Av_⫹共L兲兩u_⫹共L兲兩²兴

⫺␥

⬘

^B 4K

⬘

^共e⫺

4iK⬘^L⬘⫺1兲u_⫹共L兲²v_⫹共L兲*

册

^eiK⬘L⬘

⫹O共␧²兲. 共A4兲

2. Coefficients of the master equation

We obtained in Sec. II C a relation between f_n⫹1 and f_n, relation 共9兲. The nonlinear coefficient P involved in this equation is given by

P⫽e^{(⫺gL⫺iK⬘L⬘)}

冋

^cos␪⫺

冉

^{␸21⫹␴1␺2*␺1*2⫹A␴␺2兩␺1兩2}

⫹␴␺2兩␺2兩²

冊

^{⫹sin␪⫺e2iK⬘L⬘}

冉

^{␸22⫹␴1*␺1␺22}

⫹A␴␺1*_兩␺2兩²⫹␴␺1兩␺1兩²

冊册

^{, 共A5兲}

where

␴⫽␥

⬘

^L

⬘

^, 共A6兲

␴1⫽⫺i␥

⬘

^B 4K

⬘

^共e

4iK⬘^L⬘⫺1兲, 共A7兲

␺1⫽e^(g⫺iK)Lcos␣^sin␪_⫹⫺e^(g⫹iK)Lsin␣^cos␪_⫹^, 共A8兲

␺2⫽e^(g⫺iK)Lcos␣^cos␪_⫹⫹e^(g⫹iK)Lsin␣^sin␪_⫹^, 共A9兲

␸1⫽e^(g⫺iK)L␮1cos␣⫹e^(g⫹iK)L␮2sin␣^, 共A10兲

␸2⫽e^(g⫹iK)L␮2cos␣⫺e^(g⫺iK)L␮1sin␣^, 共A11兲

␮1⫽␥^B共e^(2g⫹4iK)L⫺1兲

g⫹2iK cos␪_⫹^sin2␪_⫹

⫹␥共e^2gL⫺1兲

g 共cos³␪_⫹⫹A cos␪_⫹^sin2␪_⫹兲, 共A12兲

␮2⫽␥^B共e^(2g⫺4iK)L⫺1兲

g⫺2iK cos²␪_⫹^sin␪_⫹

⫹␥共e^2gL⫺1兲

g 共sin³␪_⫹⫹A cos²␪_⫹^sin␪_⫹兲. 共A13兲 The coefficient Q is

Q⫽e^⫺iK⬘L⬘␾1cos␪⫺⫹e^iK⬘L⬘␾2sin␪⫺, 共A14兲

with

␾1⫽e^⫺iKLcos␣^cos␪_⫹⫹e^iKLsin␣^sin␪_⫹^, 共A15兲

␾2⫽e^iKLcos␣^sin␪_⫹⫺e^⫺iKLsin␣^cos␪_⫹^. 共A16兲 We also give the expression of 兩Q兩² used in expression 共10兲of the gain, as

兩Q兩²⫽兩␾1兩²cos²␪_⫺⫹1

2sin 2␪_⫺共e^2iK⬘L⬘␾1*␾2

⫹e^{⫺2iK⬘L⬘}␾1␾2*_兲⫹兩␾2兩²sin²␪_⫺^. 共A17兲

The coefficientsDrandDiof the master Eq.共15兲are explic- itly computed using Eq.共14兲and the above formulas.

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